Computational complexity for minimizing wire length in two- and multi-layer channel routing

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1 Advances in Computational Sciences and Technology ISSN Volume 10, Number 8 (2017) pp Research India Publications Computational complexity for minimizing wire length in two- and multi-layer channel routing Swagata Saha Sau, Sumana Bandyopadhyay and Raat Kumar Pal Department of Computer Science and Engineering, University of Calcutta, Kolkata , West Bengal, India. Abstract In this paper we have proved that two-layer (VH) no-dogleg channel routing problem of wire length minimization in absence of vertical constraints is NPcomplete. Next, we have proved that the said problem is also NP-complete for a general channel instance containing both horizontal and vertical constraints in three-layer VHV and multi-layer Vi+1Hi, 2 i<dmax no-dogleg channel routing models, where vertical and horizontal layers of interconnections alternate. We concluded that if the density of the channel (dmax) is same as the number of nets present in the channel, then the no-dogleg channel routing problem of wire length minimization in two-layer VH is polynomial time computable though channel has no vertical constraints. Keywords: Channel routing problem; No-dogleg routing; NP-complete; VLSI; Wire length minimization. INTRODUCTION The wire length minimization problem is a longstanding problem in VLSI physical design automation [4], though a very little amount of work has been addressed so far in this theme of research [4-6]. Some channel routing problem (CRP) of wire length minimization is NP-hard [2,3,4,7,8]. In most of the cases, heuristics have been designed to obtain near optimal solutions for solving hard problems, but still there are some important problems in channel routing, whose nature is yet to know, whether these are polynomial time computable or beyond polynomial time computable (or intractable) [4]. This is the prime motivation to us to consider the addressed problems.

2 2686 Swagata Saha Sau, Sumana Bandyopadhyay and Raat Kumar Pal In this paper, we primarily address the computational complexity issues of the wire length minimization problem in two-layer channel routing in the absence of vertical constraints (TNWLM). We define this problem in no-dogleg reserved layer Manhattan routing model, when only horizontal constraints are there (in a channel) among the nets with two or more terminals. Later on, we have considered the wire length minimization problem in three-layer VHV, and multi-layer Vi+1Hi, 2 i<dmax, nodogleg channel routing model for channels consisting of both horizontal and vertical constraints (MNWLM). In this paper, we have proved that the stated problems are NP-complete. Sequencing to minimize weighted completion time (SMWCT) is an established NP-complete problem [1]. First of all, we reduce an instance of SMWCT to a corresponding instance of the first problem, TNWLM (of wire length minimization in routing a channel having no vertical constraints). As it is not possible to compute a solution of an instance of SMWCT in polynomial time, then eventually we prove that the problem TNWLM is at least as hard as SMWCT. Next, we prove that this result is equally applicable to MNWLM, which is the wire length minimization problem for a general channel instance (that contains both horizontal as well as vertical constraints). Generally, a channel is a rectangular routing region that is formed when two blocks are placed sideways on a chip floor and a routing region is sandwiched in between. Terminals to be connected inside a channel are placed on the periphery of the blocks. A set of terminals that need to be electrically connected by wires is called a net. Channel is specified by net list. Here, initially we consider the reserved two-layer (VH) no-dogleg Manhattan channel routing model, where one layer is reserved for vertical wire segments and the other layer is reserved for horizontal wire segments. Later on, we have assumed multi-layer channel routing model with alternating vertical (V) and horizontal (H) layers of interconnect where both the top and bottom layers are vertical layers (and no extreme horizontal layer is allowed) [4]. Now, we formally define the two constraints present in general CRP [4]. These constraints are termed as horizontal constraints and vertical constraints. Two nets are said to be horizontally constrained, if their intervals overlap when they are assigned to the same track. Vertical constraints in a channel determine the order of placement of nets from top to bottom or vice versa. Horizontal and vertical constraints of the channel are represented by horizontal constraint graph (HNCG) and vertical constraint graph (VCG) [5]. The parametric difference of net ni is denoted by pdi and it represents the difference between the number of top terminals (TTi) and the number of bottom terminals (BTi) of the net. Decision Versions of Different Associated Problems in Channel Routing for Minimizing Wire Length: Here we pose the decision versions of associated problems

3 Computational complexity for minimizing wire length in two- and multi-layer 2687 to the work under consideration. Problem: Sequencing to minimize weighted completion time (SMWCT). Instance: Set T of tasks, partial order on T, for each task tt a length l(t) Z + and a weight w (t) Z +, and a positive integer K. Question: Is there a one processor schedule σ for T that obeys the precedence constraints and for which the sum of (σ (t) + l(t)).w(t) is K or less for all t T? Problem: Two-layer no-dogleg wire length minimization in channel routing in the absence of vertical constraint (TNWLM). Instance: A channel specification of n multi-terminal nets that do not contain any vertical constraint whose channel density is dmax (< n), and a positive integer K. Question: Does there exist a two-layer no-dogleg routing solution such that the total wire length is Kor less? Problem: Wire length minimization in multi-layer Vi+1Hi, 2 i<dmax, no-dogleg general channel routing (MNWLM). Instance: A general channel specification of n multi-terminal nets for their assignment in the multi-layer Vi+1Hi, 2 i < dmax, routing model, where dmax < n is the channel density, and a positive integer K. Question: Does there exist a multi-layer Vi+1Hi, 2 i< dmax, no-dogleg routing solution so that the total wire length is K or less? We may note that SMWCT is NP-complete in the strong sense and remains so even if all task lengths are 1 or all task weights are also 1 [1]. If the partial order is replaced by individual task deadline, the resulting problem remains NP-complete in the strong sense, but can be solved in polynomial time if all task weights are equal [1]. If there are individual task release times instead of deadlines, the resulting problem remains NP-complete in the strong sense, even if all task weights are same as 1 [1]. NP-completeness of Wire Length Minimization in Two-layer Channel Routing in the Absence of Vertical Constraints, Three-layer VHV and Multi-layer Vi+1Hi, 2 i <dmax No-dogleg General Channel Routing: Here we shall first prove TNWLM is NPcomplete by reducing an instance of SMWCT to the corresponding instance of this problem in polynomial time.

4 2688 Swagata Saha Sau, Sumana Bandyopadhyay and Raat Kumar Pal Theorem 1: TNWLM is NP-complete. Proof: To proof this Theorem, we first show that TNWLM NP. Given a guessed twolayer no-dogleg routing solution S of some channel instance I without having any vertical constraint, we can verify in polynomial time whether S is a valid routing solution of the given channel instance I, where total wire length required is less than an positive integer K. Therefore, TNWLM NP. Here we establish a polynomial transformation over an instance I of the well-known NP-complete problem SMWCT to a corresponding instance of TNWLM. The transformation is as follows. Let us assume that each task has same weight and is equal to 1, i.e., w(t) = 1 for all t T. Hence, according to problem SMWCT, ( t) l( t). w( t) is less than or equal to K. Now as w(t) = 1, therefore, ( t) l( t) K. tt We presume that in a channel a set Nof nets contains n multi-terminal nets that do not contain any vertical constraints, where each net ni, 1 i n, has pdi the parametric difference of ni, which is the difference between top terminals (TTi) and bottom terminals (BTi), and l(ni) is the vertical wire length required to connect the top and bottom terminals of net ni. To show that TNWLM is NP-complete we consider the following transformation from SMWCT to TNWLM. For all tasks t T, of SMWCT is transformed into a net ni of TNWLM (ni N). Partial order < on T, for associated pair of tasks in T is represented as horizontal nonconstraint among the corresponding nets in TNWLM. For one processor schedule σ for T that obeys the precedence constraints and for which we have corresponding pdi of net ni N). Here σ(ti) > σ(t) signifies that pdi > pd. We assign net ni either above or to the same track of net n if they do not have horizontal constraint. l(t) is the length of task t T. We represent l(t) as total vertical wire length of net ni N (i.e., l(ni)). Now, we construct the net list (or channel specification) I of TNWLM which is reduced from an instance I of SMWCT. We construct the channel specification in such a way that the channel does not have vertical constraints. As we discussed earlier that the partial orders among tasks t T of instance I of SMWCT represent horizontal nonconstraints among associated nets of instance I of TNWLM, and σ(t) of t represents the parametric difference pdi of net ni. We have written below the steps of the transformation that eventually produces a channel instance without any vertical constraint, where an HNCG (G = (V, E)) and pdi of each net ni are obtained by one-to- tt

5 Computational complexity for minimizing wire length in two- and multi-layer 2689 one correspondence from instance I of SMWCT. The subsequent steps of constructing I are as follows. 1. Construct HCG (G = (V, E)) [4] from HNCG (G = (V, E)) by complementing G. 2. Start from a simplicial vertex vi in G (that corresponds to net ni) having highest pdi. 3. Insert pdi + 1 leftmost columns to the channel under construction and assign terminals of net ni to the top net list in each of these columns where the bottom terminal in each associated column contains a nonterminal (i.e.,'0'). 4. For a vertex v (corresponds to net n) with {v, vi} E that has highest pd among the vertices whose corresponding nets are yet to assign but form a maximal clique including vi (or have horizontal constraint with ni), insert pd + 1 columns to the right of the rightmost column inserted up to this point in time (into channel under construction) and assign terminals of net n to the top net list in each of these columns where the bottom terminal in each associated column contains a non-terminal (i.e., '0'). 5. As for all adacent vertices vi corresponding terminals get connected to the top of the channel after insertion of each of the associated nets to the channel under construction, a column is appended to the right of the rightmost column that containing a non-terminal to the top and a terminal of net ni corresponding to vertex vi to the bottom. 6. Delete vertex vi from G 7. Continue Step 2 through Step 6 until set V of G gets empty. Now, consider a (desired) routing solution of minimum wire length as follows for which we do the following. Now, once the channel specification is completely obtained, we can employ MCC1 [4] with necessary modifications as follows. Here instead of assigning a net to the topmost track that starts from the leftmost terminal of the net list, net ni with highest pdi among the nets having horizontal constraints, gets more privilege to be assigned to a track nearer the top than the remaining ones. Thus, in this way, we find track Xi, where net ni is to be placed (for i =1, 2,..., n) and the total number of tracks required to route the nets becomes TN. We may assume H as the total horizontal wire needed

6 2690 Swagata Saha Sau, Sumana Bandyopadhyay and Raat Kumar Pal to route all the nets and si being the set of nets assigned to track Xi. Hence, the total wire length required for routing is as follows: X itti TN X i 1 BTi H. Now,for SMWCT, t l t. w t K or t l t K, where w t 1, for all t. 1 TN is tt tt As stated earlier, for TNWLM, the aim is to minimize the total vertical wire length along with horizontal wire segment essential for a valid routing. Thus, the expression needed to be optimized as written below. 1 TN is X TT TN X 1 BT i i i i H X TT TN X 1 BT t t 1 TN is i i i i i i H X TT TN X 1 BT t TT BT 1 TN is i i i i i i i H ' K pdi H K, where1 TN 1 TN is Here, pd ' i must be less than or equal to, hence, K H 1 TN i S is a positive integer. On the opposite side, starting from SMWCT, t l t wt wt l t i i i., where is one.hence, replacing by 1 i n 1 i n vertical wire length, we get the following expression. X itti TN X i 1 BTi ti 1 TN is 1 TN is X itti TN X i 1 BTi pdi 1 TN is 1 TN is ' K H pdi K 1 TN is We can easily prove that the time and space consumed by the reduction process from I of SWMCT to I of TNWLM and vice-versa is bounded by some polynomial function of number of tasks existing is an instance of SWMCT. If the density of the channel dmax is n, i.e., the number of nets present in the channel of

7 Computational complexity for minimizing wire length in two- and multi-layer 2691 two-layer VH no-dogleg wire length minimization in channel routing in absence of vertical constraint, we need n number of tracks to route the nets to the channel because all the nets present in the channel are horizontally constraint to each other. Hence, we can get minimum wire length routing solution in polynomial time only if we place the nets in different tracks according to their parametric differences. In similar way we can prove that the wire length minimization problem in three-layer VHV and multi-layer no dogleg general channel routing MNWLM (Vi+1Hi, 2 i <dmax) is NP-complete by reducing the instance of an NP-complete problem such as SMWCT to the instance of three-layer VHV and the instance of MNWLM (Vi+1Hi, 2 i < dmax), in polynomial time, respectively. CONCLUSIONS We showed that the decision version of two-layer VH no-dogleg wire length minimization in channel routing (with dmax<n) in the absence of vertical constraints, three-layer VHV and multi-layer Vi+1Hi, 2 i < dmax no-dogleg wire length minimization in general channel routing problem are NP-complete by transforming the instance of the known NP-complete problem, i.e. sequencing to minimize weighted completion time problem to the instance of each problem in polynomial time. We also concluded that if the density of the channel dmax is n, i.e. the number of nets present in the channel then the two-layer VH no-dogleg channel routing problem for wire length minimization is polynomial time computable though channel has no vertical constraints. REFERENCES [1] Garey M. R., Johnson D. S.: Computers and Intractability: A Guide to the Theory of NP Completeness (W. H. Freeman and Company. San Francisco. New York 1979). [2] Greenberg R., Jaa J., Krishnamurthy S.: On the difficulty of Manhattan channel routing, Journal of Information Processing Letters, 1992, 44, pp [3] LaPaugh A. S.: Algorithms for integrated circuit layout: An analytic approach, (Ph.D. Thesis, Massachusetts Institute of Technology Cambridge, MA, USA, 1980). [4] Pal R. K.: Multi-Layer Channel Routing Complexity and Algorithms (Narosa Publishing House, New Delhi, 2000). [5] Saha Sau S., Pal R. K.: A re-router for optimizing wire length in two-and four-layer no-dogleg channel routing, Proceedings of the 18th International

8 2692 Swagata Saha Sau, Sumana Bandyopadhyay and Raat Kumar Pal Symposium on VLSI Design and Test, 2014, doi: /ISVDAT [6] Saha Sau S., Pal R. K.: An Efficient Algorithm for Reducing Wire Length in Three- Layer Channel Routing, in: Chaki R. et al. (Eds.). Series Advances in Intelligent Systems and Computing, Pp Print ISBN: , Online ISBN: , 2015, doi: / Publisher Springer India. [7] Schaper G. A.: Multi-layer channel routing (Ph. D. Thesis, Department of Computer Science, University of Central Florida, Orlando, 1989). [8] Szymanski T. G.: Dogleg channel routing is NP-complete, IEEE Transaction on CAD of Integrated Circuits and Systems, 1985, 4, pp